Super Lie-infinity algebra of closed superstring field theory?

Question

Bosonic closed string field theory is famously governed by a Lie n-algebra for $n = \infty$ whose $k$-ary bracket is given by the genus-0 (k+1)-point function in the BRST complex of the string.

One might therefore expect that, analogously, closed superstring field theory (in any of its variants) is governed by a lift of that to a super Lie n-algebra for $n = \infty$.

The closest to an identification of such that I am aware of is in

• Yuji Okawa, Barton Zwiebach, Heterotic String Field Theory (arXiv:hep-th/0406212)

where substructures of the bosonic string field $L_\infty$-algebra are paired with the super-ingredients. This seems to go in the right direction, but does not quite identify a super $L_\infty$-algebra structure.

Is there, meanwhile, anything more known that may complete the picture here?

This post imported from StackExchange Physics at 2014-03-03 18:43 (UCT), posted by SE-user Urs Schreiber

Comments

Why do you use both "$L_\infty$-algebra" and "Lie $n$-algebra for $n = \infty$", when the article you linked indicates that they are the same thing? Is there a subtle distinction I am missing?

This post imported from StackExchange Physics at 2014-03-03 18:43 (UCT), posted by SE-user Scott Carnahan

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There is indeed no distinction, and that's what I wanted to implicitly emphasize a little, with an eye towards the BLG "3-algebra" excitement ncatlab.org/nlab/show/BLG+model#3AlgebraStructure.

This post imported from StackExchange Physics at 2014-03-03 18:43 (UCT), posted by SE-user Urs Schreiber

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Superstring field theory actually works differently than this "straightforward" generalization of bosonic string field theory.

This post imported from StackExchange Physics at 2014-03-03 18:43 (UCT), posted by SE-user Luboš Motl

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@Dilaton I have reshowed that one and hidden the other, because this one has Lubos's comment on it, which is not there on the other one. 

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Could you make the attritution for the post red? It's very confusing to see the attributes for comments in red, yet the attributes for the post in blue. Using red is a clear way of showing what's imported, and users need to know this quickly to decide if something unique has been posted here..

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@physicsnewbie No, I won't. I can't. Font colours are removed when a post is edited. Too bad, but sorry, no. This isn't even legally required. It's quite obvious to me even when it is not in read; the font size is smaller, the text is the same, the links are the same.  

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If there are technical issues, of course they have the last word, but I think "physicsnewbie" above has a point: while I do agree that it is a good idea to import old posts, it would be good to make it more easily visible that they are imported, or at least that they are _old_, otherwise it can get a bit confusing. I for one was taken a bit by surprise to suddenly see myself apparently having posted a question here, which I didn't even remember having posted...

Answer 1

A while after I had asked the above question back in October 2011 (on another forum, the question now having been imported here), the following article appeared, which claims to produce what I was wondering about:

• Branislav Jurčo, Korbinian Muenster, Type II Superstring Field Theory: Geometric Approach and Operadic Description, JHEP 1304:126 (2013) arXiv/1303.2323 doi

based on

• Chungsheng James Yeh, Topics in superstring theory, PhD thesis, Berkeley 1993 (SPIRE)

 Their introduction has a useful survey. First they recall the traditional way of going about superstring field theory:

 The first attempt towards a field theory of superstrings was initiated by the work of Witten

• Edward Witten, Interacting field theory of open superstrings, Nuclear Physics B, Volume 276, Issue 2 (1986)

by seeking a Chern-Simons like action for open superstrings similar to the one of open bosonic string field theory (Witten 86). The major obstacle compared to the bosonic string is the necessity of picture changing operators?. Indeed, the cubic superstring theory of (Witten 86a) turns out to be inconsistent due to singularities arising form the collision of picture changing operators

• C. Wendt, Scattering amplitudes and contact interactions in Witten’s superstring field theory, Nuclear Physics B, Volume 314, Issue 1.

In order to circumvent this problem, another approach was pursued which sets the string field into a different picture

• C.R. Preitschopf, C.B. Thorn, S. Yost, Superstring field theory Nuclear Physics B, Volume 337, Issue 2.

• I.Ya. Aref’eva, P.B. Medvedev, A.P. Zubarev, New representation for string field solves the consistency problem for open superstring field theory, Nuclear Physics B, Volume 341, Issue 2.

but upon including the Ramond sector, the modified superstring field theory suffers from similar inconsistencies

• M. Kroyter, Superstring field theory equivalence: Ramond sector, Journal of High Energy Physics, Volume 2009, Issue 10.

These two approaches are based on the small Hilbert space, the state space including the reparametrization ghosts and superghosts as they arise from gaugefixing. Upon bosonization of the superghosts, an additional zero mode arises which allows the formulation of a WZW like action for the NS sector of open superstring field theory

• Nathan Berkovits, Super-Poincare Invariant Superstring Field Theory, (hep-th/9503099)

In contrast to bosonic string field theory, BV quantization of this theory is more intricate than simply relaxing the ghost number constraint for the fields of the classical action

• Nathan Berkovits, Constrained BV description of string field theory, Journal of High Energy Physics, Volume 2012, Issue 3.

• M. Kroyter, Y. Okawa, M. Schnabl, S. Torii, Barton Zwiebach, Open superstring eld theory I: gauge xing, ghost structure, and propagator, Journal of High Energy Physics, Volume 2012, Issue 3.

Finally, there is a formulation of open superstring field theory that differs from all other approaches in not fixing the picture of classical fields

• M. Kroyter, Superstring field theory in the democratic picture, Advances in Theoretical and Mathematical Physics, Volume 15, Number 3.

Then they come to the "geometric" approch which I was asking about whether it has been generalized from the bosonic to the superstring:

On the other hand, the construction of bosonic closed string field theory (Zwiebach 92) takes its origin in the moduli space of closed Riemann surfaces. Vertices represent a subspace of the moduli space, such that the moduli space decomposes uniquely into vertices and graphs,and do not apriori require a background. Graphs are constructed from the vertices by sewing together punctures along prescribed local coordinates around the punctures. But an assignment of local coordinates around the punctures, globally on the moduli space, is possible only up to rotations. This fact implies the level matching condition and via gauge invariance also the (b_0^-)-constraint.

In an almost unnoticed work (Yeh), the geometric approach developed in bosonic closed string field theory, as described in the previous paragraph, has been generalized to the context of superstring field theory. Neveu-Schwarz punctures behave quite similar to punctures in the bosonic case, but a Ramond puncture describes a divisor on a super Riemann surface rather than a point. As a consequence, local coordinates around Ramond punctures, globally defined over super moduli space, can be fixed only up to rotations and translation in the Ramond divisor.

A given background provides forms on super moduli space

• A. Belopolsky, New Geometrical Approach to Superstrings, hep-th/9703183.

• L. Alvarez-Gaume, P. Nelson, C. Gomez, G. Sierra, C. Vafa, Fermionic strings in the operator formalism, Nuclear Physics B, Volume 311, Issue 2.

in the sense of geometric integration theory on supermanifolds, and in particular the geometric meaning of picture changing operators has been clarified

• Alexander Belopolsky, Picture changing operators in supergeometry and superstring theory(arXiv:hep-th/9706033).

Integrating along an odd direction in moduli space inevitably generates a picture changing operator. Thus, the ambiguity of defining local coordinates around Ramond punctures produces a picture changing operator associated with the vector field generating translations in the Ramond divisor. The bpz inner product plus the additional insertions originating from the sewing define the symplectic form relevant for BV quantization. As in the bosonic case, we require that the symplectic form has to be non-degenerate, but the fact that the picture changing operator present in the Ramond sector has a non-trivial kernel, forces to impose additional restrictions besides the level matching and b_0^- = 0 constraint on the state space. The purpose of (Jurco-Muenster 13) is to describe the construction of type II superstring field theory in the geometric approach.

To the extent that this is correct, it means that also closed superstring field theory has an action functional that is a Chern-Simons-type term for a (super) Lie n-algebra (for n = infinity, a "strong homotopy Lie algebra"), that's equation (4.12) of (Jurco-Muenster 13).

Comments

Thanks for the update Urs! That is a very interesting story.

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Dear Urs,

What is the difference between an L_\infty and super L_\infty algebra, in your understanding? For the superstring I would have thought that we would only need to upgrade the vector space to include  (left and rightmoving) NS and R states, but the algebraic structure would be the same.

BTW, the paper of Jurco-Muenster has some serious shortcomings. The construction of quantum superstring field theory is still very much an open problem.

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A super L-infinity algebra is to an L-infinity algebra as a super-Lie algebra is to a Lie algebra: in addition to the \(\mathbb{Z}\)-grading of the homotopy structure there is a super-\(\mathbb{Z}/2\mathbb{Z}\)-grading (as in arXiv:1308.5264),

Regarding Jurco-Muenster: I guess that's true, yes.

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Dear Urs,

I am somewhat confused about sign issues and am wondering whether "super L_\infty" is really the right word for the kind of algebraic structure one expects for superstrings.

The difference between Lie algebras and super-Lie algebras seems clear. But L_\infty by itself already seems like a "super Lie algebra," since the vector space on which the multilinear maps act already has a Z_2 grading representing "even" and "odd" elements. At least this is what happens in the context of SFT.

For the closed bosonic string the physical field is always even. One might naively think that for the superstring the NS-NS and R-R fields should be even, while the fermionic R-NS and NS-R states should be odd. But I think that all sectors should be even,  otherwise symmetry of the closed string products would forbid nontrivial couplings with fermions. In fact I think this is consistent with the conventions of Jurco-Muenster. 

The algebra of closed superstring fields should have three Z-gradings, representing ghost number and leftmoving and rightmoving picture number, and two Z_2 gradings, the first distinguishing NS and R states and the second counting Grassmann parity. Only the last grading, Grassmann parity, seems to play a role in the definition of the L_\infty structure. But Grassmann parity is already there for the bosonic string.

This is why I am confused as to the meaning of super-L_\infty in this context. Probably this is just a result of my ignorance of the received terminology on these matters, but I would like to understand if there is something I'm missing.

Best,

Ted

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Dear Ted,

okay, thanks for digging into this. There are maybe three issues to be separated:

1. What is a super L-infinity algebra?
2. Should it appear in super string field theory where a plain L-infinity algebra appears in bosonic string field theory?
3. What happens in Jurco-Muenster?

Regarding 3. I suppose you are right, by their comments on "+- orientation" the odd-odd signs which I am expecting to see don't actually arise in their article. Hm.

Regarding 2: Do we agree that the BRST complex for the superstring has an \(\mathbb{N} \times (\mathbb{Z}/2\mathbb{Z})\)-bigrading, given by (ghost number | super-grading)? 

It frequently happens that super-sign rules in the further presence of cohomological signs become confusing. For this purpose I like to stick with Deligne-Freed's Sign manifesto, where the key example is the bigrading on super-differential forms (on the last pages there). 

This bigrading models the bigrading also on super-BRST complexes. And in particular...

Regarding 1: Where a degreewise finite-dimensional L-infinity algebra/algebroid is dually a dg-algebra on an \(\mathbb{N}\)-graded symmetric algebra with archetypical example being the de Rham complex of a manifold (corresponding to the tangent Lie algebroid), so a degreewise finite-dimensional super-L-infinity algebra/algebroid is dually a super-dg-algebra on an \(\mathbb{N}\times (\mathbb{Z}/2\mathbb{Z})\)-graded module with archetypical example being the super-de Rham complex of a supermanifold (corresponding to the super tangent Lie algebroid), whose bigrading I take to be as in Deligne-Freed's Sign manifesto,

Dualizing this super-dg-algebra definition of finite-type super-L-infinity algebras one finds the bracket-version of the definition which generalizes to infinite-dimensional super L-infinity algebras as they would appear in super string field theory.

If they appear there. Now, i am not claiming that I understand whether they appear there, after all, in the question which starts the above thread I am asking if this turns out true.

You seem to say that it won't turn out true. I am prepared to accept this, but I am not sure yet if I understand why. In bosonic SFT the L-infinity structure is on the BRST complex of the bosonic string. Therefore (maybe naively) I expect that in super string field theory there should be a super L-infinity algebra structure on the super-BRST complex of the superstring. 

(Maybe to sort this out we should switch to email,if you still have the energy.)

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Dear Urs,

Thank you for your reply. I must admit that I am not certain I understood Jurco-Muenster's conventions. However, I know that in the context of the open string several authors take the Ramond field to have the same Grassmann parity as the NS field. Then it seems logical that all four closed string sectors should likewise share the same Grassmann parity.

But it is unclear to me how "Grassmann parity" as I'm calling it relates to what Deligne-Freed call "cohomological degree" and "parity." I must say that I did not understand the possibility that the closed string state space could have a \(\mathbb{Z}\times\mathbb{Z}_2\) bigrading, if means that anticommutation of states produces seperately a sign from the \(\mathbb{Z}\) and a sign from the \(\mathbb{Z}_2\). Let me write a few equations to make sure we agree on the implications of this bigrading. It would imply that closed string products have the symmetry

\(b_n(...,A,B,...) = (-1)^{A_1B_1+A_2B_2}b_n(...,B,A,...)\)

where the subscripts in the exponents refer to the cohomological degree and parity. The Jacobi identity would take the form

\(b_2(b_2(A,B),C)+(-1)^{C_1(A_1+B_1)+C_2(A_2+B_2)} b_2(b_2(C,A),B) \\ +(-1)^{A_1(B_1+C_1)+A_2(B_2+C_2)} b_2(b_2(B,C),A)=0. \)

In addition, I would assume that the string products have odd cohomological degree but even parity.

With this kind of structure we have two possibilities: the Ramond states have even cohomological degree and even parity, or that the Ramond states have odd cohomological degree and odd parity. Otherwise symmetry of the closed string products would be inconsistent with Ramond couplings. The signs associated to these two possibilities will in general be different, since in the first case the Ramond states commute with the multi-string products, and in the second case they will anticommute.

I would have thought that we should follow the first possibility for Ramond states, since this seems to agree with what has been done for the open string. In this case the second parity grading seems algebraically irrelevant for SFT. But perhaps we should follow the second possibility. Or perhaps these assignments are equivalent in the end.

Feel free to email me if you want to continue the conversation elsewhere... (my email address can be found on my papers). I replied here to take advantage of TeX equations.

Best

Ted

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Hi Ted,

First of all: yes, so the point of that sign rule which I have in mind is to introduce -- on top of all other signs that are there -- a minus sign whenever two super-odd things are interchanged. Regarding your formulas for the \(b_n\): depending on what exactly \(b_n\) is meant to be, there might be more signs coming from just the fact that this is a skew bracket, if I understand your notation well, but anyway and regardless of that, the point is that there should be the additional sign which you write \((-1)^{A_2 B_2}\)on top of whatever other signs there are already from the non-susy story.

Now regarding the superstring sectors: both the chiral R-sector and the NS-sector are chiral super-Virasoro algebras in themselves (hence in particular have underlying them super-vector spaces), with both super-even elements and super-odd elements. So I would not assign a uniform super-grading to the whole of one of these sectors.

For definiteness, just to agree on what we are talking about without doubt, let's look at some text that displays the standard formulas, say... this one here: http://www.maths.ed.ac.uk/~jmf/Research/PVBLICATIONS/nsr.pdf . We see super-even elements \(\ell_n\) of the NS-sector in (3.1) there and super-odd elements \(g_n\)in (3.2). Similarly for the R-sector there are even elements \(\ell_n\) as before and super-odd elements \(f_n\) in (4.1). (If you are thinking of 0-modes, of course, in the NS-sector there are no super-odd 0-modes.) And of course for the closed string all this is doubled.

In particular for instance the actual RR-fields in target space terminology are pairs of two super-odd zero-modes in the RR-sector and hence are super-even in total. But the actual target space spinors in R-NS and NS-R are pairs of a super-even and a super-odd guy and hence are super-odd, as befits a spinor. 

Accordingly, I would say that there must be an extra sign whenever two spinors are interchanged, as always, and the sign rule that I am talking about does give that.

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Dear Urs,

Thank you for discussing this. Though it is easy to get lost in signs I am making progress understanding the fundamental issue.

I understand that it seems natural that for superstrings we should need an extra "supersign" in the \(L_\infty\) structure. But let me explain why I am still skeptical.

In the usual conventions for worldsheet computations, there is no bigrading---worldsheet fields are either Grassmann even or Grassmann odd. For example, the c-ghost is taken to anticommute with the worldsheet fermion. Perhaps this convention is not natural from other points of view, but I am hesitant to abandon it, and will take it as given.

Now one can ask which Grasmann parity is carried by a chiral Ramond vertex operator. Since this involves spin fields which alone don't have well-defined Grassmann parity, there may be some ambiguity here. The only useful discussion I'm aware of appears on page 2 of Michishita http://arxiv.org/abs/hep-th/0412215. The upshot is that the indefiniteness of Grassmann parity cancels between the spin field and the superconformal ghosts. Let me assume that the Grassman parity assignment Michishita makes to the chiral Ramond vertex operator is correct.

It follows that the vertex operators for the R-NS and NS-R sectors of the closed superstring must be Grassmann odd. Since R-NS and NS-R states represent spacetime fermions, these vertex operators should be multiplied by an anti-commuting spinor field. Let me call it \(\psi^\alpha\).

Now it seems there is a chance for bigrading to come into play. The question is: does \(\psi^\alpha\) commute or anticommute with the worldsheet fields? More topically, which of the following equations is true?

\(Q(\psi^\alpha|\Omega_\alpha\rangle) = -\psi^\alpha Q|\Omega_\alpha\rangle?\\ Q(\psi^\alpha|\Omega_\alpha\rangle) = \psi^\alpha Q|\Omega_\alpha\rangle ?\)

If the first equation is true, then there is really only a single relevant grading, which should be identified with worldsheet Grassmann parity plus \(\psi^\alpha\)number (mod \(\mathbb{Z}_2\)). All four sectors of the closed superstring field would carry even parity, and the theory would be described by an ordinary (non-super) \(L_\infty\) structure.

If the second equation is true, then we really have a bigrading and a super-\(L_\infty\) structure. In this case, I would identify "cohomological degree" with worldsheet Grassmann parity, and "super-parity" with \(\psi^\alpha\) number (mod \(\mathbb{Z}_2\)). Then the NS-R and the R-NS fields have odd cohomological degree and odd super-parity.

I have been assuming that the first equation is true, since this is the convention followed by Michishita (and seemingly others), and in BV quantization of the bosonic string one also typically takes the anticommuting spacetime fields to anticommute with anticommuting worldsheet fields.

Right now, however, I cannot exclude the second possibility which leads to a super \(L_\infty\) structure. Without thinking too carefully it seems that both choices are consistent. This is confusing, since I would think that the difference between super and non-super \(L_\infty\) would be dictated by consistency, and would not be a matter of convention.

Best,

Ted

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Thanks, we are progressing. Thanks for having the patience and accuracy for this kind of issue, this is good.

So, indeed, in addition to the "internalization sign rule" which I have been highlighting above (which is Deligne-Freed's "relentless application of the sign rule" from their Sign manifesto), there is the "super odd sign rule". Where in the former case one has a \(\mathbb{Z} \times \mathbb{Z}_2\)-bigrading, in the latter case this is fused to just a single \(\mathbb{Z}_2\)-grading, the "total parity" or whatever one wants to call it.

The super odd sign rule is popular among authors who like to base everything on supergeometry and suppress the homotopy-theoretic aspects -- probably actually the majority of authors.

Now here is one fact that I am aware of: if one has a super-(BV-)BRST complex, then while its interpretation as the Chevalley-Eilenberg algebra of an L-infinity algebra in super vector spaces only works for the relentless Deligne-Freed bigrading, it's cohomology is in fact invariant under change of the two sign rules. For the case of just plain super Lie algebras this is discussed here, but the argument directly generalizes.

So I suppose this will account for the bulk of the issue at hand: on cohomology, hence in "physical gauge invariant sectors" both sign conventions give the same answer.

However, for full super string field theory we need more than just the cohomology of the BV-BRST complex. I would suspect that only with the relentless Deligne-Freed sign rule a proper L-infinity algebraic description of super string field theory will be possible. 

But given that super string field theory does not exist much, yet, either way, maybe that may only be decided by actually doing it.

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Dear Urs,

Thank you for answering my questions.

With either the "internalization sign rule" or the "super-odd sign rule," I believe one should be able to write down a gauge invariant action which will reproduce the correct on-shell amplitudes.

In this sense, the super-\(L_\infty\) structure which one might have expected does not seem so well motivated from "essential" string field theory considerations. I am still trying to understand your statement that "a super-(BV-)BRST complex is the Chevalley-Eilenberg algebra of an L-infinity algebra," and what relevance this would have for string field theory. So there may still be other considerations which would motivate the bigrading of the closed string field.

I will keep both conventions in mind!

Best,

Ted